Rotors that mesh and cannot drive each other

The second shape is not a choice: give one body a profile and a motion relative to another, and the shape the other has to be is the envelope of the first, computed rather than designed. Any shape has a partner pushed that as far as it goes, handing the same routine a flank invented to be nothing in particular and getting back a mate that held the ratio exactly.

A Roots blower asks a question neither of those did. Its two rotors turn on parallel shafts at the same speed in opposite senses, each with two or three lobes, and they are the same part: one is the other, turned. So the mate of the shape is required to be the shape again. Whether such a shape exists, how much of it is still a choice, and whether the pair can then do what a pair of gears does are three separate questions. The first two have tidy answers. The third has an answer that surprises people who know the first two.

Half of every lobe is given, and half is computed #

Each rotor has a pitch circle of radius 50, and the two pitch circles roll on each other at the point midway between the shafts. A lobe has two halves: a tip outside the pitch circle and a root inside it, and a rotor carries its tips and roots alternately round its circumference. When a tip of one rotor sweeps through the line of centres, it sits in a root of the other.

That arrangement makes the construction almost trivial to state. Choose a tip. Run the meshing routine with that tip on the first rotor and the second rotor turning the other way at the same speed, over the half pitch the tip spends in mesh, and it returns the shape the second rotor needs where the tip goes, which is a root. Now put that root on the first rotor as well, between its own tips.

The first rotor’s tips fit the second rotor’s roots by construction. The second rotor’s tips, which are the first rotor’s tips turned, fit the first rotor’s roots, because each root was computed as the shape a tip of exactly that form needs, and conjugacy runs both ways: the partner of a partner is the original. So a rotor that is its own mate is not found by a search. Any tip that leaves the pitch circle and comes back to it at the two ends of its half pitch gives one, and the only design decision in the whole lobe is the tip.

The classical choice is a tip traced by a small circle rolling round the outside of the pitch circle, an epicycloid arch. For the arch to fill exactly half a lobe pitch, the rolling circle’s circumference has to equal that arc, which fixes its radius at the pitch radius over twice the lobe count: 12.5 for two lobes, 8.33 for three. The tip’s height is twice that. The same circle rolling round the inside of the pitch circle traces a hypocycloid, and textbooks draw the root as that curve. The computation knows nothing of either circle. It is given only the tip.

The 479 computed points lie within 3.7 × 10⁻¹⁴ of the hypocycloid for two lobes and 3.2 × 10⁻¹⁴ for three, which is the arithmetic’s last digits. Every one of them is a genuine contact. At every sampled position but the two hand-overs at the ends of the half pitch, the tip has exactly one contact with its mate, and the meshing equation holds there to 5 × 10⁻⁹.

That agreement is two routes to one shape. The first is a velocity condition, solved at each instant and mapped into a moving frame, with no circle anywhere in it. The second is a circle rolling inside another. They agree for the reason the cycloidal tooth form works at all: a describing circle rolling between two pitch circles traces conjugate curves on both of them at once, because the point it carries always lies on a line through the pitch point. On a Roots rotor the two pitch circles are the same size, so the curves traced on the two rotors are the same curves. The rotor is its own mate for the classical reason as well as the computed one.

The tip is a choice, up to a height #

The cycloidal tip is a convention rather than a requirement, and a circular tip shows it. Take an arc through the two half-pitch points on the pitch circle, rising to a chosen height at its middle, and ask for its root.

A circular tip 20 high gets a perfectly good root. It is not a circular arc, and nothing about it is simple to write down; its tightest bend has a radius of 36.9. But it is a boundary. Its contacts run steadily along it from one end to the other, and a rotor made of those tips and those roots meshes with its twin at every position, exactly as the cycloidal rotor does.

Make the arc taller and the root it demands bends harder. At 36 high it has stopped being a curve a solid could have: its contacts reverse direction 102 times as the tip passes through. That is what an envelope that has turned back on itself looks like when it is sampled, a locus of points each of which satisfies the meshing equation and which together bound nothing. The mate a tall circular tip needs would have to be cut away by the tip itself, just as a rack cutter takes back flank it generated a moment earlier.

Measured against height, the tightest bend falls steadily and reaches zero at a fold. For two lobes the fold is at a height of 32.00, found by bisection to a bracket two thousandths wide; for three lobes it is at 23.20. A tip slightly shorter than the fold has a root that is a boundary, and a tip slightly taller does not.

The cycloidal tips sit well inside both limits, 25 against 32 for two lobes and 16.7 against 23.2 for three. So the classical lobe is not the tallest lobe that works, and the height is worth having: in a housing that clears the tips, a taller lobe sweeps more air round per turn. What limits the height is neither the pitch circle nor the neighbouring lobe. It is the radius of curvature of the root, and the root is there to receive the mate’s tip. The constraint exists only because the mate is the same part.

Which way the contact pushes #

None of that has asked whether the pair can do work. Two gears in mesh can: one shaft is driven, the other is loaded, and the contact carries torque across. The natural assumption is that two exactly conjugate rotors can too, since they touch at every instant and hold their ratio exactly.

The contact can only push, along its common normal and out of the rotor doing the pushing. Its turning effect on the other rotor is the push times the normal’s moment arm about the other shaft. Because the normal passes through the pitch point, that arm has a simple form: the pitch radius times the component of the normal’s direction square to the line of centres. When the normal is square to the line of centres the arm is the whole pitch radius, which is where the contact is momentarily rolling. When the normal lies along the line of centres it passes through both shafts and the arm is zero. That second position is a dead centre in everything but name: the contact can push as hard as it likes and turn neither rotor.

The measured arm for cycloidal lobes runs from zero up to 50, down through zero to −50 and back, with zeros at −90°, 0° and 90°: every time a tip lies on the line of centres, whichever rotor the tip belongs to. At each zero the arm changes sign. For half of every lobe pitch, the only contact there is pushes the driven rotor backwards. Counted over the 359 positions with one contact, the contact turns the mate forward at 50.4% of them, a half to the resolution of the sampling.

The circular tip 20 high does no better. Its arm is smaller everywhere, peaking at 34.6 rather than 50, so it needs a larger push for the same torque even where it works, and it passes through zero at the same three angles and changes sign at each.

Three positions make it concrete. At −30° the normal leans one way across the line of centres and the push turns the mate forward with an arm of 43.3. At 0° the tip is in the bottom of the mate’s root, the normal is the line of centres, and it runs straight through both shafts. At 30° the normal leans the other way, and the same push turns the mate backward with an arm of −43.3. Nothing has broken. The contact is still there, the ratio is still exact, and the direction in which the contact can deliver torque has reversed.

The describing circle predicts the arm #

For the cycloidal lobe the curve has a closed form, and it is a second route to the measurement. A contact between cycloidal curves always sits on the describing circle, which touches both pitch circles at the pitch point, so the common normal is the chord from the pitch point to the contact. When a tip is on the line of centres, the contact is at the far end of the circle’s diameter and the chord lies along the line.

As the rotors turn through an angle φ, the describing circle rolls along the pitch circle by the same arc, which turns it through 2n·φ about its own centre because its radius is the pitch radius over 2n. A chord from a fixed point of a circle turns through half the angle its far end moves round the centre, so the normal turns through n·φ away from the line of centres. Its component square to the line is sin nφ, and the arm about the mate’s shaft is −R·sin nφ.

Against that formula the computed arm agrees to 8.4 × 10⁻⁹ for two lobes and 1.2 × 10⁻⁸ for three, at every sampled position. The measurement knows nothing of describing circles; it found each contact by solving the meshing equation on the tips of both rotors and took the moment of the normal about a shaft. The formula knows nothing of contacts. Two routes, one curve, and the zeros are where the sine says they are.

No tip escapes it #

The formula belongs to the cycloidal tip, but the zeros belong to every tip. A tip that is symmetric about its own centre line has its highest point on that line, and there its surface is square to its radius. When that point is in contact, the contact normal is its radius and also has to pass through the pitch point, so the point is on the line of centres and the normal passes through both shafts. As the tip passes through, the contact moves from one side of its highest point to the other, and the normal swings from one side of the line of centres to the other, so the arm changes sign. The same happens at the bottom of each root, which is the mate’s highest point in contact.

A rotor that stays in contact with its twin all the way round passes every point of its profile through contact, the highest point of each tip included. So a self-mating pair in continuous contact reaches a dead centre 2n times a turn, four for two lobes and six for three, and between them the contact pushes forward and backward in turn. With symmetric lobes the two directions take equal shares. No choice of tip changes that, because choosing the tip is choosing only what happens between the zeros: how large the arm grows, and whether it is the sine of the cycloidal lobe or the flatter curve of the circular one.

A cam meets the same condition from the other side, since a cam is a conjugate pair with a follower in place of a second rotor. Its push fails when the contact normal turns square to the follower’s line of motion, a pressure angle of 90°, and a cam designer keeps well away from that by choosing a large enough base radius. A Roots pair has no such choice, because its dead centres are forced by the symmetry of the part and not by its size.

Why a Roots blower’s rotors never touch #

That is why the rotors of a Roots blower do not drive each other and are not meant to touch. The shafts are coupled by a pair of ordinary timing gears outside the housing, which carry the torque from the driven shaft to the other through involute teeth whose line of action never passes through both shafts. The rotors are made slightly smaller than their computed shapes, so that they clear each other everywhere, and the clearance is a seal rather than a contact: it restricts air leaking back from outlet to inlet without carrying any load.

A pair made to touch, with no timing gears, could push its mate forward only in the quarter turns where the arm is positive. Through the quarter turns between, the driven rotor would have to be carried by its own momentum, and the contact would either let go or push it back.

Seen this way, the computed shape is doing a different job from a gear tooth’s. It is not a way of transmitting motion; the timing gears do that. It is the shape that lets two rotors turn past each other with nothing between them but a narrow gap at every position, and the gap can be narrow everywhere only because the shapes are conjugate. Rotors that were not conjugate could still be made to clear each other, but only with a gap that opens as they turn, and the air leaks through the widest gap on the cycle. It is the job the corners of a rotary engine’s rotor do against their housing, and in both machines the geometry is exact so that a seal can be thin.

The clearance also removes a second problem. A blower whose rotors both touched and were timed by gears would have two independent paths fixing the relative angle of its shafts, which is overconstraint in exactly the linkage sense: both paths would share the load only if they agreed to the micron, and otherwise one of them would carry it all or bind. The gap gives the whole job of timing to the gears and leaves the rotors the job of sealing, which they can do even when they are not quite right.

Planar sections, circular tips, symmetric lobes #

The rotors are planar sections. A straight Roots rotor is a planar section extruded along the shaft, so everything here applies along its whole length. Rotors whose lobes twist along the shaft put the tip on the line of centres at different angles at different sections, and whether the arm summed along the length can avoid a zero was not measured.

The fold heights belong to circular tips on these pitch circles. A tip of another shape folds at another height, and the tallest tip of any shape was not searched for.

The equal shares depend on symmetric lobes. A lobe that is not symmetric about its centre line still has a highest point and still reaches a zero there, but its forward and backward intervals need not be equal, and no asymmetric self-mating rotor was computed.

Clearance and leakage are not modelled. The gap is described, not sized, and nothing here says how much air it passes.

What comes next: the clearance that is the seal #

The clearance that is the seal. Shrinking both conjugate rotors by the same small offset leaves a gap between them, and whether its smallest width stays constant through the turn, and how much it varies for a pair that is not conjugate, is a direct measurement on the rotors computed here. It would turn the claim that conjugacy makes a thin seal possible into a number.

Twisted lobes. A lobe that twists along the shaft puts different sections at a dead centre at different moments. The torque the whole rotor can pass is the sum along its length, and whether a twist of one lobe pitch or less makes that sum keep one sign all the way round is a question the arm computed here can answer section by section.

Unlike rotors that do drive each other. The two rotors of a screw compressor have different lobe counts and asymmetric profiles, and in oil-flooded machines one drives the other with no timing gears at all. What the asymmetry buys, and whether a pair of unlike rotors can keep its moment arm of one sign all the way round, is the question for shapes that mesh with something other than themselves; the lobes of a cycloidal disc, which mesh with pins and do carry load, are the nearest case already computed.